Fracture and Complexity: One Century since Griffith’s Milestone

Description

The book explores the two opposite natural trends of composite systems: (i) order and structure emerging from heterogeneity and randomness, and (ii) instability and chaos arising from simple nonlinear rules. Providing insights into the rapidly growing field of complexity sciences, the book focuses on the role of complexity in fracture mechanics. It firstly discusses the occurrence of self-similarity and fractal patterns in deformation, damage, fracture,  and fragmentation of heterogeneous materials and the apparent scaling of the nominal mechanical properties of disordered materials, as well as of the time-to-failure after fatigue and creep loading. Then the book addresses criticality in the acoustic emissions from damaged structures and tectonic faults. Further, it examines the snap-back instability in the structural behavior of relatively large composite structures in the framework of catastrophe theory, and lastly describes the transition toward chaos in the dynamics of cracked elements. Preface Contents Author and Contributors About the Author Previous Authored or Edited Books Contributors Part IFrom Stress Singularity to Strain Energy Release Rate: Local Versus Global Approach to Fracture Mechanics 1 Stress Concentration at the Notch Root 1.1 Preliminary Remarks 1.2 Plane Stress Condition 1.3 Plane Strain Condition 1.4 Thick-Walled Cylinder 1.5 Circular Hole in a Plate Subjected to Tension 1.6 Analytical Functions 1.7 Kolosoff–Muskhelishvili Method 1.8 Elliptical Hole in a Plate Subjected to Tension 1.9 Griffith’s Fracture Energy Criterion 1.10 Experimental Confirmations References 2 Stress Intensification at the Crack Tip 2.1 Preliminary Remarks 2.2 Westergaard’s Method 2.3 Mode II and Mixed Modes 2.4 Stability of Crack Propagation 2.5 Elasto-Plastic Material 2.6 Isoparametric Finite Elements 2.7 Quarter Point Element 2.8 Degenerate Triangular Element 2.9 Estimation of Error in the Numerical Evaluation of Stress-Intensity Factors 2.10 Experimental Determination of the Critical Stress-Intensity Factor KIC for Metallic Materials (ASTM E399 Standard) 2.10.1 Characteristics and size of Test Specimens 2.10.2 Test procedure 2.10.3 Testing Apparatus 2.10.4 Analysis of the Load–Displacement Diagrams 2.11 Determination of the Fracture Toughness of Rocks 2.11.1 Calculation of the Fracture Toughness KCB 2.11.2 Correction of Fracture Toughness for Nonlinearity References 3 Stress Intensification at the Vertex of a Re-entrant Corner 3.1 Preliminary Remarks 3.2 Singular Stress Field in the Case of Linear Elastic Material 3.3 Singular Stress Field in the Case of Strain-Hardening Material 3.4 Extension of the Plastic Zone Around a Re-entrant Corner 3.5 Generalized Fracture Toughness 3.6 Critical Amplitude of a Re-entrant Corner 3.7 Size-scale Effects in Structural Elements with Re-entrant Corners 3.8 The Notch Blunting Effect References 4 Energy Approach to Fracture Mechanics 4.1 Preliminary Remarks 4.2 Relation Between Energy and Stress-Singularity Treatments: Irwin’s Theorem 4.3 Local Compliance of a Cracked Structural Element 4.4 J-Integral 4.4.1 Independence of the J-Integral from the Integration Curve 4.4.2 Variations of Energy 4.4.3 Identity of the J-Integral with the Strain Energy Release Rate mathcalG I 4.5 Experimental Investigations 4.5.1 Experimental Determination of Fracture Toughness Parameters 4.5.2 Comparison Between the Fracture Parameters Obtained 4.6 Experimental Determination of Fracture Energy for Mortar and Concrete (Rilem Recommendation) 4.6.1 The Proposed Testing Method 4.6.2 Testing Procedure and Characteristics of the Test Specimens References 5 Mixed-Mode Crack Propagation 5.1 Preliminary Remarks 5.2 Criterion of Maximum Hoop Stress 5.3 Criterion of Minimum Strain Energy Density 5.4 Criterion of Maximum Released Energy 5.5 J-Vector Criterion 5.6 Experimental Tests and Empirical Criteria 5.7 Scale Effects in Relation to Crack Size 5.8 Effect of Stress Parallel to the Crack 5.9 Plastic Effects at the Crack Tip 5.10 Directional Stability in Crack Propagation 5.11 Loci of Resistance in the Principal Stress Plane 5.11.1 Mohr–Coulomb Criterion 5.11.2 Griffith’s Macroscopic Criterion 5.11.3 Friction on Griffith Cracks 5.11.4 Microcrack Population Model References Part IIFrom Simple Nonlinear Constitutive Laws to Complex Mechanical Behaviour: Catastrophe and Chaos 6 Nonlinear Crack Models 6.1 Preliminary Remarks 6.2 Plastic Zone at the Crack Tip 6.3 Strain Energy Density Criterion—Strain-Hardening Materials 6.3.1 Material Behaviour 6.3.2 Isotropic Versus Kinematic Hardening 6.3.3 Effect of Loading Step 6.4 Strain Energy Density Criterion—Strain-Softening Materials 6.4.1 Material Behaviour 6.4.2 Variation in the σ-ε Softening Slope 6.4.3 Effect of Loading Step 6.4.4 Size Effects on Strength and Ductility 6.4.5 Centre-Cracked Slab in Tension 6.4.6 Three-Point Bending of a Reinforced Beam with Edge Crack 6.4.7 Eccentric Compression of Wall with Edge Crack 6.5 Cohesive Crack Model—Mode I 6.5.1 Localized Strain 6.5.2 Three-Point Bending Test 6.5.3 Numerical Procedure 6.6 Ductile–Brittle Transition and Snap-Back Instability 6.6.1 Influence of Initial Crack Depth and Beam Slenderness 6.7 Cohesive Crack Model—Mixed Mode 6.7.1 Experimental Program 6.7.2 Directional Stability of the Crack Trajectory 6.8 Loss of Symmetry and Bifurcations 6.8.1 Crack Length Control Scheme 6.8.2 Solution of the Single Crack Growth Step 6.9 Nonlinear Crack Concepts Applied to Compression: The Overlapping Crack Model 6.10 Overlapping Crack Model for Eccentric Compression 6.10.1 Numerical Algorithm 6.10.2 Comparison Between Model Predictions and Experimental Results 6.10.3 Size-Scale and Slenderness Effects in Eccentric Compression Tests References 7 Size-Scale Transition from Ductile to Brittle Failure 7.1 Preliminary Remarks 7.2 Dimensional Analysis 7.2.1 Buckingham’s Theorem 7.3 Different Structural Geometries 7.4 Size-Scale Effects on Apparent Fracture Toughness 7.4.1 Metals 7.4.2 Concrete and Rocks 7.4.3 Cohesive Crack Model 7.4.4 Influence of the Shape of the Cohesive Diagram σ–w 7.4.5 Damage Model Versus Cohesive Model 7.5 Dugdale Plastic Zone Correction 7.6 BCS Crack Model 7.7 Virtual Crack Propagation Model 7.8 Cohesive Limit Analysis 7.8.1 Uniaxial Tensile Loading of Slabs 7.8.2 Three-Point Bending of Beams [34, 35] 7.8.3 Three-Point Bending of Deep Beams 7.9 Size-Scale Effects on Apparent Bending Strength 7.10 Brittleness Limit for Infinite Size-Scale 7.11 Structural Response Versus Crack Growth Resistance Curve 7.11.1 Scale Effect on the Structural Response 7.11.2 Strain-Hardening Material 7.11.3 Linear-Elastic Material 7.11.4 Three-Point Bending Geometry 7.11.5 Scale Effect on the J-Resistance Curve References 8 Mechanical Behaviour of Reinforced Structural Elements 8.1 Preliminary Remarks 8.2 Crack Growth Stability in Steel-Bar Reinforced Concrete Elements: Rotation Compatibility Condition 8.2.1 Statically Indeterminate Reaction of Reinforcement 8.2.2 Bending Moment of Reinforcement Plastic Flow 8.2.3 Rigid-Hardening Behaviour of the Cracked Beam Section 8.2.4 Bending Moment of Matrix Fracture 8.2.5 Stability of the Process of Matrix Fracture and Steel Plastic Flow 8.2.6 Summary 8.3 Crack Growth Stability in Steel-Bar Reinforced Concrete Elements: Crack Opening Displacement Compatibility Condition 8.3.1 Displacement Compatibility Condition and Statically Indeterminate Reaction of Reinforcement 8.3.2 Combined Stress-Intensity Factor 8.3.3 Crack Propagation 8.3.4 Moment Versus Rotation Response 8.3.5 Comparison with Experimental Results 8.3.6 Experimental Confirmation of Snap-Back Behaviour 8.3.7 Concluding Remarks 8.4 Crack Growth Stability in Fibrous Composites: Discrete Model 8.4.1 Theoretical Model 8.4.2 Displacement Compatibility Conditions 8.4.3 Crack Propagation 8.4.4 Structural Response of the Cracked Element 8.4.5 Two Fibres 8.4.6 Large Number of Fibres 8.4.7 Concluding Remarks 8.5 Crack Growth Stability in Fibrous Composites: Continuous Model 8.5.1 Continuous Model 8.5.2 Discrete Model Versus Continuous Model 8.5.3 Continuous Model Versus Experimental Results 8.5.4 Bridging Option Versus Cohesive Option 8.5.5 Concluding Remarks 8.6 Hysteretic Behaviour of Steel-Bar Reinforced Concrete Elements: Rotation Compatibility Condition 8.6.1 Elastic–Plastic Shake-Down Under Repeated Loadings 8.6.2 Critical Crack Depth and Bending Moment 8.6.3 Fatigue Crack Growth and Energy Dissipation 8.7 Hysteretic Behaviour of Steel-Bar Reinforced Concrete Elements: Crack Opening Displacement Compatibility Condition 8.7.1 Moment Versus Rotation Diagrams 8.7.2 Beam A 8.7.3 Beam B 8.7.4 Beam C 8.7.5 Experimental Comparisons 8.7.6 Concluding Remarks 8.8 Hysteretic Behaviour of Fibrous Composites 8.8.1 Concluding Remarks 8.9 Transitions of Reinforced Concrete Beams in Flexure: Tensile, Shearing, Crushing Failures 8.9.1 Modelling Flexural and Shear Cracks 8.9.2 Modelling Concrete Crushing 8.9.3 Transition Between Different Failure Modes 8.9.4 Experimental Evidences 8.10 Cohesive/Overlapping Crack Model for Nonlinear Analysis of Reinforced Concrete Beams 8.10.1 Mathematical Formulation 8.10.2 Numerical Algorithm 8.10.3 Computation of the Elastic Coefficients 8.10.4 Parametric Analysis and Experimental Comparisons 8.10.5 Size-Scale Effects 8.10.6 Effect of the Tensile Steel Reinforcement Percentage 8.10.7 Effect of the Steel Reinforcement in Compression 8.10.8 Effect of the Concrete Compressive Strength 8.10.9 Effect of the Stirrups Confinement 8.11 Lower and Upper Reinforcement Limits to the Ductile Behaviour of Concrete Members: Minimum Reinforcement and Rotational Capacity 8.11.1 Minimum Reinforcement 8.11.2 Models for Computing Minimum Reinforcement 8.11.3 Application of Dimensional Analysis to Lightly RC Members 8.11.4 Comparison of Predictions and Experimental Results 8.11.5 Parametric Analysis and Discussion 8.11.6 Plastic Rotation Capacity 8.11.7 Concluding Remarks References 9 Debonding and Decohesion at the Interface Between Dissimilar Media 9.1 Preliminary Remarks 9.2 Stress Singularities at Multi-material Interfaces 9.3 Generalized Stress-Intensity Factors and Computational Methods 9.4 Interface Crack Propagation Criteria 9.5 Nonlinear Interface Constitutive Laws and Interface-Contact Elements 9.6 Delamination of Plated Beams 9.7 Fibre-Matrix Debonding in Microstructured Composites References 10 Nonlinear and Chaotic Behaviour in the Vibration of Cracked Bodies 10.1 Preliminary Remarks 10.2 Theoretical Continuum Approach 10.3 Theoretical Discrete Approach 10.4 Period Doubling Cascade 10.5 Parametrical Simulations 10.6 General Discussion References Part IIIFrom Complex Morphological Patterns to Simple Mechanical Models: Fractality and Critical Phenomena 11 Fractality of Scale-Invariant Cohesive Constitutive Laws 11.1 Preliminary Remarks 11.2 Weibull’s Theory 11.2.1 The Weakest-Link Concept 11.2.2 Safety Factor 11.3 Defect Size Distribution of Self-similarity 11.3.1 Weibull Parameters 11.3.2 Experimental Results 11.4 Fractal Geometry 11.4.1 Hausdorff Dimension 11.4.2 Box-Counting Dimension 11.4.3 Random Fractals 11.5 The Fractal (Scale-invariant) Cohesive Crack Model 11.5.1 The Effect of Microstructural Disorder 11.5.2 Scale-Invariant Cohesive Crack Model 11.5.3 Comparison with Experimental Data 11.6 Dimensional Transition from Order to Disorder: Multi-fractal Scaling Laws 11.6.1 Multi-fractal Scaling Laws 11.6.2 Comparison with Experimental Data 11.7 Fractal Overlapping Crack Model 11.8 Rotation Versus Curvature Fractal Scaling in Bending Failure 11.9 New Tools for New Challenges References 12 Fractional Calculus Applied to Fractal Media and Nonlocal Continua 12.1 Preliminary Remarks 12.2 Fractional Calculus: A Brief Review 12.3 Fractional Calculus and Fractal Functions: The Local Fractional Derivative 12.4 Kinematic and Static Equations for Fractal Media 12.5 The Fractal Bar 12.6 Eringen’s Approach and Nonlocal Fractional Elasticity 12.7 The Nonlocal Fractional Elastic Bar 12.8 Numerical Simulations References 13 Scaling and Fractality in Sub-critical Fatigue and Creep Crack Growth 13.1 Preliminary Remarks 13.2 Analytical Correlations Between the Fatigue Properties of Engineering Materials 13.3 Generalized Cumulative Fatigue Damage Formulation 13.4 Generalized Fatigue Crack Propagation Formulation 13.5 Interpretation of Specimen-Size Effects on Paris’ Law According to Incomplete Self-similarity 13.6 Interpretation of Crack-Size Effects on Paris’ Law According to Fractal Geometry 13.7 Interpretation of Specimen-Size Effects on Wöhler’s Curve According to Fractal Geometry 13.8 Fatigue Propagation of Cracks Subjected to Mixed-Mode Loading 13.9 Scaling Laws for Creep Deformation and Rupture Time 13.10 Specimen-Size Effects on Creep Rupture Time 13.11 Crack-Size Effects on Creep Crack Growth References 14 Critical Phenomena and Acoustic Emission in Structural Elements and the Earth’s Crust 14.1 Preliminary Remarks 14.2 Critical Phenomena in Disordered Materials 14.3 Scaling Laws in Geophysics and Mechanics 14.4 Acoustic Emission and Detection of Crack Evolution in Damaging Structures 14.4.1 Dissipated and Emitted Energies 14.5 AE Waves and Signal Waves 14.5.1 Measuring System 14.5.2 Signal Processing 14.5.3 Event Counting 14.5.4 Ring-Down Counting 14.5.5 Parameter-Based AE Techniques 14.5.6 Signal-Based AE Techniques 14.5.7 Spectral Analysis 14.5.8 Source Localization 14.5.9 Amplitude Distribution Analysis 14.5.10 Moment Tensor Analysis 14.5.11 Cracking Modes and Typical AE Signals 14.5.12 AE Localization Procedures 14.6 Size-scale Effects in AE Monitoring 14.6.1 Experimental Assessment 14.7 Time-scale Effects in AE Monitoring 14.7.1 Experimental Assessment 14.8 AE Frequency–Magnitude Statistics and b-Value Analysis 14.9 Fractal Dimension Evolution of Microcrack Networks in Disordered Materials 14.9.1 Self-similar and Self-affine Crack-Size Distributions 14.10 Experimental Observations on AE 14.10.1 In-situ Retrofitted RC Beam Test 14.10.2 Three-Point Bending Test 14.10.3 Concrete Specimen in Compression 14.11 From Criticality to Final Collapse: Evolution of the Fractal Dimension D and of the b-Value 14.11.1 Different Approaches for Damage Domain Characterization in Disordered Materials: Fractal Energy Density and b-Value Statistics 14.12 Acoustic and Electromagnetic Emissions from Quasi-brittle Materials 14.12.1 Models for EME 14.13 Experimental Observations on AE and EME 14.13.1 AE and EME Measurements 14.13.2 Test Results 14.14 Regional Seismicity and AE Structural Monitoring 14.14.1 From Acoustic Emission to Earthquakes 14.15 b-Value Analysis on Medieval Towers as a Seismic Precursor 14.15.1 Damage Detection in the Towers 14.15.2 The b-Value Analysis 14.15.3 Correlation Between AE Activity in the Towers and Regional Seismicity 14.15.4 Fractal Dimensions from Space–Time Correlation Integral 14.16 Universality in Fracture Systems 14.16.1 Gutenberg–Richter Law and Scale Invariance in the Timing of Ruptures 14.16.2 Scaling Laws for Waiting-Time Distributions of Italian Seismicity 14.16.3 Scaling Laws for Waiting-Time Distributions in Concrete Fracture 14.16.4 Discussion on the Results 14.17 Statistical Seismic Precursors and Preparation Time for the Latest Earthquakes in Italy 14.17.1 Methods and Data Set 14.17.2 The L’Aquila Earthquake 14.17.3 The Emilia Earthquake 14.17.4 The Amatrice–Norcia Earthquake 14.18 Fracto-Emissions as Seismic Precursors 14.18.1 Acoustic, Electromagnetic and Neutron Emissions: The Crack-Size Evolution in the Earthquake Preparation Zone 14.18.2 The Case Study of “San Pietro–Prato Nuovo” Gypsum Mine Station References 15 Scaling and Fractality in Fragmentation and Comminution: Applications to Drilling and Wear 15.1 Preliminary Remarks 15.2 Fractal Fragmentation Theory of Quasi-Brittle Materials in Compression 15.3 Energy Dissipation Under Fragmentation 15.4 Fractal Scaling Laws 15.4.1 Shape Effects 15.4.2 Size Effects 15.4.3 Geometrical Multi-fractal Extension 15.5 One-, Two-, and Three-Dimensional Universal Laws for Fragmentation Due to Impact and Explosion 15.5.1 Three-Dimensional Theory 15.5.2 Two-Dimensional Theory 15.5.3 One-Dimensional Theory 15.6 An Example of Application: The Asteroid Collision 15.7 Fractal Comminution Approach to Evaluate the Drilling Energy Dissipation 15.7.1 Experimental Fractal Assessment: The Single-Scratch Test 15.7.2 Experiments on Drilling Comminution 15.8 Power Balance for Drilling Comminution 15.9 Coupled Fractal Theory of Drilling and Wear 15.9.1 Coupled Law of Drilling and Wear Velocities 15.9.2 Fractal Coupled Law of Wear and Drilling Velocities 15.10 Evolutionary Fractal Theory of Erosion and Experimental Assessment on MIR Space Station 15.10.1 Classical Erosion 15.10.2 Classical Coupled Erosion 15.10.3 Fractal Erosion 15.10.4 Fractal Coupled Erosion 15.10.5 Experimental Comparison with In-Flight Erosion Due to Space Debris Impacts on MIR Orbital Space Station References